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n-S(n), n+S(n) are both perfect squares

Source: 2023 Czech-Polish-Slovak Match Junior, Team p1 CPSJ

May 5, 2024
number theoryPerfect SquaresPerfect Squaresum of digits

Problem Statement

Let S(n)S(n) denote the sum of all digits of natural number nn. Determine all natural numbers nn for which both numbers n+S(n)n + S(n) and nāˆ’S(n)n - S(n) are square powers of non-zero integers.
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